An Introduction toRiemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
- To understand the notion of homotopy of paths in a topological space
- To understand concatenation of paths in a topological space
- To sketch how the set of fixed-end-point (FEP) homotopy classes of loops at a point
becomes a group under concatenation, called the First FundamentalGroup
- To look at examples of fundamental groups of some common topological spaces
- To realise that the fundamental group is an algebraic invariant of topological spaces which
helps in distinguishing non-isomorphic topological spaces
- To realise that a first classification of Riemann surfaces can be done based on their fundamental
groups by appealing to the theory of covering spaces
Keywords:
Path or arc in a topological space, initial or starting point and terminal or ending point of a path,
path as a map, geometric path, parametrisation of a geometric path,
homotopy, continuous deformation of maps, product topology, equivalence of
paths under homotopy, fixed-end-point (FEP) homotopy, concatenation of paths,
constant path, binary operation, associative binary operation, identity element for
a binary operation, inverse of an element under a binary operation, first fundamental group, topological invariant

This lecture introduces the fundamental group of a surface. We begin by discussing when two paths on a surface are homotopic, then defining multiplication of paths, and then multiplication of equivalence classes or types of loops based at a fixed point of the surface.
The fundamental groups of the disk and circle are described.
This is part of a beginner's course on Algebraic Topology given by N J Wildberger at UNSW.
A screenshot PDF which includes AlgTop21 to 29 can be found at my WildEgg website here: http://www.wildegg.com/store/p119/product-Algebraic-Topology-C-screenshot-pdf

published:11 Nov 2011

views:16951

An unsolved conjecture, the inscribed square problem, and a clever topological solution to a weaker version of the question, the inscribed rectangle problem (Proof due to H. Vaughan, 1977), that shows how the torus and mobius strip naturally arise in mathematical ponderings.
Patreon: https://www.patreon.com/3blue1brown
Get 10% your domain name purchace from https://www.hover.com/, by using the promo code TOPOLOGY.
Special shout out to the following patrons: Dave Nicponski, Juan Batiz-Benet, Loo Yu Jun, Tom, Othman Alikhan, Markus Persson, Joseph John Cox, Achille Brighton, Kirk Werklund, Luc Ritchie, RiptaPasay, PatrickJMT , Felipe Diniz, Chris, Andrew Mcnab, Matt Parlmer, NaokiOrai, Dan Davison, JoseOscar Mur-Miranda, Aidan Boneham, BrentKennedy, Henry Reich, Sean Bibby, PaulConstantine, Justin Clark, Mohannad Elhamod, Denis, Ben Granger, Ali Yahya, Jeffrey Herman, and Jacob Young
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC-
Various social media stuffs:
Twitter: https://twitter.com/3Blue1Brown
Facebook: https://www.facebook.com/3blue1brown/
Reddit: https://www.reddit.com/r/3Blue1Brown

published:04 Nov 2016

views:602753

I will explain a viewpoint on resurgence via Floer-type topology of Lefschetz thimbles in the complexified path integral. In particular, so called 2d-4d wall-crossing formalism of Gaiotto, Moore and Neitzke arises naturally in this framework, without any reference to supersymmetric field theories.

Jose Bravo demonstrates how to navigate through the QRM topology. Next he demonstrates the Configuration Monitor's ability to display the configurations of the network firewalls and routers in a vendor neutral way and to maintain a historical record of the configuration changes. He demonstrates how to highlight the differences between two sets of firewall configurations. He also demonstrate's QRM's ability to compare configurations between two different firewalls or any two devices.
Next, Jose demonstrates how QRM discovers new devices in the environment and he demonstrates how to search for devices in the topology so that missing machines can be located.
Jose also demonstrates QRM's capability to identify the path routes between two different machines and how the path between the two machines may be blocked.
IBM Owner: CalvinPowers

published:09 Oct 2014

views:3359

https://pitp2015.ias.edu/

published:12 Aug 2015

views:8146

Explanation for the article: http://www.geeksforgeeks.org/topological-sorting/
This video is contributed by Illuminati.

Recording of JonathanWilliamson's talk at the 2012BlenderConference on how to work with complex topology on organic and hard-surface models.
The video recording is pretty rough, the lighting in the room made it very difficult to get good footage. Thanks to the kind user that filmed the talk for us!

Path (topology)

The initial point of the path is f(0) and the terminal point is f(1). One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path. Note that a path is not just a subset of X which "looks like" a curve, it also includes a parameterization. For example, the maps f(x) = x and g(x) = x2 represent two different paths from 0 to 1 on the real line.

A loop in a space X based at x ∈ X is a path from x to x. A loop may be equally well regarded as a map f: I → X with f(0) = f(1) or as a continuous map from the unit circleS1 to X

This is because S1 may be regarded as a quotient of I under the identification 0 ∼ 1. The set of all loops in X forms a space called the loop space of X.

A topological space for which there exists a path connecting any two points is said to be path-connected. Any space may be broken up into a set of path-connected components. The set of path-connected components of a space X is often denoted π0(X);.

Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important topological properties include connectedness and compactness.

Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs (Greek-Latin for "geometry of place") and analysis situs (Greek-Latin for "picking apart of place"). Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics.

Topological sorting

In the field of computer science, a topological sort (sometimes abbreviated toposort) or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge uv from vertex u to vertex v, u comes before v in the ordering. For instance, the vertices of the graph may represent tasks to be performed, and the edges may represent constraints that one task must be performed before another; in this application, a topological ordering is just a valid sequence for the tasks. A topological ordering is possible if and only if the graph has no directed cycles, that is, if it is a directed acyclic graph (DAG). Any DAG has at least one topological ordering, and algorithms are known for constructing a topological ordering of any DAG in linear time.

Examples

The canonical application of topological sorting (topological order) is in scheduling a sequence of jobs or tasks based on their dependencies; topological sorting algorithms were first studied in the early 1960s in the context of the PERT technique for scheduling in project management(Jarnagin 1960). The jobs are represented by vertices, and there is an edge from x to y if job x must be completed before job y can be started (for example, when washing clothes, the washing machine must finish before we put the clothes to dry). Then, a topological sort gives an order in which to perform the jobs.

Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that if a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

This has two important specializations: 1) If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem). 2) The image of a continuous function over an interval is itself an interval.

Motivation

This captures an intuitive property of continuous functions: given f continuous on [1, 2] with the known values f(1) = 3 and f(2) = 5. Then the graph of y = f(x) must pass through the horizontal line y = 4 while x moves from 1 to 2. It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting your pencil from the paper.

Theorem

The intermediate value theorem states the following.

Consider an interval in the real numbers and a continuous function . Then,

Mod-02 Lec-08 Homotopy and the First Fundamental Group

An Introduction toRiemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
- To understand the notion of homotopy of paths in a topological space
- To understand concatenation of paths in a topological space
- To sketch how the set of fixed-end-point (FEP) homotopy classes of loops at a point
becomes a group under concatenation, called the First FundamentalGroup
- To look at examples of fundamental groups of some common topological spaces
- To realise that the fundamental group is an algebraic invariant of topological spaces which
helps in distinguishing non-isomorphic topological spaces
- To realise that a first classification of Riemann surfaces can be done based on their fundamental
groups by appealing to the theory of covering spaces
Keywords:
Path or arc in a topological space, initial or starting point and terminal or ending point of a path,
path as a map, geometric path, parametrisation of a geometric path,
homotopy, continuous deformation of maps, product topology, equivalence of
paths under homotopy, fixed-end-point (FEP) homotopy, concatenation of paths,
constant path, binary operation, associative binary operation, identity element for
a binary operation, inverse of an element under a binary operation, first fundamental group, topological invariant

AlgTop24: The fundamental group

This lecture introduces the fundamental group of a surface. We begin by discussing when two paths on a surface are homotopic, then defining multiplication of paths, and then multiplication of equivalence classes or types of loops based at a fixed point of the surface.
The fundamental groups of the disk and circle are described.
This is part of a beginner's course on Algebraic Topology given by N J Wildberger at UNSW.
A screenshot PDF which includes AlgTop21 to 29 can be found at my WildEgg website here: http://www.wildegg.com/store/p119/product-Algebraic-Topology-C-screenshot-pdf

18:16

Who cares about topology? (Inscribed rectangle problem)

Who cares about topology? (Inscribed rectangle problem)

Who cares about topology? (Inscribed rectangle problem)

An unsolved conjecture, the inscribed square problem, and a clever topological solution to a weaker version of the question, the inscribed rectangle problem (Proof due to H. Vaughan, 1977), that shows how the torus and mobius strip naturally arise in mathematical ponderings.
Patreon: https://www.patreon.com/3blue1brown
Get 10% your domain name purchace from https://www.hover.com/, by using the promo code TOPOLOGY.
Special shout out to the following patrons: Dave Nicponski, Juan Batiz-Benet, Loo Yu Jun, Tom, Othman Alikhan, Markus Persson, Joseph John Cox, Achille Brighton, Kirk Werklund, Luc Ritchie, RiptaPasay, PatrickJMT , Felipe Diniz, Chris, Andrew Mcnab, Matt Parlmer, NaokiOrai, Dan Davison, JoseOscar Mur-Miranda, Aidan Boneham, BrentKennedy, Henry Reich, Sean Bibby, PaulConstantine, Justin Clark, Mohannad Elhamod, Denis, Ben Granger, Ali Yahya, Jeffrey Herman, and Jacob Young
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC-
Various social media stuffs:
Twitter: https://twitter.com/3Blue1Brown
Facebook: https://www.facebook.com/3blue1brown/
Reddit: https://www.reddit.com/r/3Blue1Brown

I will explain a viewpoint on resurgence via Floer-type topology of Lefschetz thimbles in the complexified path integral. In particular, so called 2d-4d wall-crossing formalism of Gaiotto, Moore and Neitzke arises naturally in this framework, without any reference to supersymmetric field theories.

Jose Bravo demonstrates how to navigate through the QRM topology. Next he demonstrates the Configuration Monitor's ability to display the configurations of the network firewalls and routers in a vendor neutral way and to maintain a historical record of the configuration changes. He demonstrates how to highlight the differences between two sets of firewall configurations. He also demonstrate's QRM's ability to compare configurations between two different firewalls or any two devices.
Next, Jose demonstrates how QRM discovers new devices in the environment and he demonstrates how to search for devices in the topology so that missing machines can be located.
Jose also demonstrates QRM's capability to identify the path routes between two different machines and how the path between the two machines may be blocked.
IBM Owner: CalvinPowers

Recording of JonathanWilliamson's talk at the 2012BlenderConference on how to work with complex topology on organic and hard-surface models.
The video recording is pretty rough, the lighting in the room made it very difficult to get good footage. Thanks to the kind user that filmed the talk for us!

5:39

Network Theory Topology

Network Theory Topology

Network Theory Topology

Follow along with the course eBook: https://goo.gl/hhChUL
See the full course: http://complexitylabs.io/courses
We call the overall structure to a network its topology where topology simply means the way in which constituent parts are interrelated or arranged. To illustrate how network topology affects the system we look at a number of simple networks each containing the same amount of nodes but each having a different overall topology owing to the way they were connected with these topologies affecting different features to the network, such as the shortest path length or how we might control the flow of information on the network.
Twitter: https://goo.gl/Nu6Qap
Facebook: https://goo.gl/ggxGMT
LinkedIn:https://goo.gl/3v1vwF
Transcription Except:
In the previous module, we were discussing local network metrics that referred to a single node in a graph, asking how connected it was or how influential and central it was while also building up our basic vocabulary from graph theory. In the coming set of lectures we are going to be looking at global metrics that refer to the whole of graph. A we have previously noted networks are a very informal type of structure they often simple develop without any overall top down design. Someone builds a protocol for two computers to exchange information over a network and shares it with a colleague, other people see the utility of it and connects to this little networks and then more people as the network grow, until 25 years later we have a massive network of networks that is the internet.
No one planned the internet just as one really planed the global financial networks that have emerged over the past few decades, traders, investor and institutions set up connections wherever they thought there was a viable return on investment, but now that these networks are here there overall markup feeds back to effect us the user, networks may start out quite random but they often develop into some stable overall structure and understanding the patterns to this overall structure is of central importance in network theory and the focus of this section to the course.
We can call this overall structure to a network its topology, where topology simply means the way in which constituent parts are interrelated or arranged:
Within the context of a network it defines the way different nodes are placed and interconnected with each other and the overall patterns that emerge out of this.
To illustrate this further here are a set of simple networks each containing the same amount of nodes but each having a different overall topology owing to the way it is connected. We can see how these different network topologies would in turn have very different features and properties to them. Imagine they where different transportation systems in which you are trying to get form A to B. In the star topology it would only ever take you two hops to reach your destination but in the ring it might take 3, the tree structure possibly 4, and this same influence from the topology would apply if we were trying to rout water through a hydraulic network or electricity on a power grid.
This network could also represent the flow of information within a national society we can see how the centralized star structure or the tree structure would be much easier for some political regime to control and influence as opposed to the more decentralized fully connected or ring network. The point I are trying to make clear is simply that networks have overall structure and this overall topology to the network matters as it feedback to affect the actions and capabilities of the nodes on the local level.

12:16

Graph Topological Sort Using Depth-First Search

Graph Topological Sort Using Depth-First Search

Graph Topological Sort Using Depth-First Search

In this video tutorial, you will learn how to do a topological sort on a directed acyclic graph (DAG), i.e. arrange vertices in a sequence according to dependency constraints shown by edges. The sort is done by using the depth-first search algorithm as the basis, and modifying it to label vertices with sequence numbers. The complete code is available at https://www.dropbox.com/s/71fe4ttpj27jxa8/Graph.java?dl=0
and the examples graphs illustrated in the video are at https://www.dropbox.com/s/6s8i9ji1wla28we/topsort_graph1.txt?dl=0
and
https://www.dropbox.com/s/tasix4uihrs1n5p/topsort_graph2.txt?dl=0
Error: At around 6:24, the graph example used has a cycle including X, C, D, P. The edge from X to C should go in the opposite direction, from C to X. Everything else works as it should.

16:47

ArcMap 45 - عملية تصحيح الاخطاء الهندسية Topology

ArcMap 45 - عملية تصحيح الاخطاء الهندسية Topology

ArcMap 45 - عملية تصحيح الاخطاء الهندسية Topology

2:21

Coverings of the Circle

Coverings of the Circle

Coverings of the Circle

A covering of a topological space X is a topological space Y together with a continuous surjective map from X to Y that is locally bi-continuos.
The infinite spiral is for example a covering of the circle. Notice how every path on the circle can be lifted to the spiral.
If a covering has a trivial fundamental group, i.e. it does not admit any non trivial closed paths it is called the universal covering. Here we see how a closed path on the circle is lifed to a non closed path on the spiral. Indeed the infinite spiral is the universal covering of the circle.
The name universal comes from an other property: The universal covering is a covering of every other covering. This is shown here with the 2:1 and the 3:1 covering of the circle. The universal covering covers both of them, but the 3:1 does not cover the 2:1.
From this universality property it follows also that every topological space has a unique universal covering. (not shown)
This Video was produces for a topology seminar at the LeibnizUniversitaet Hannover.
http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1

In this lecture we formalize the idea of a continuous mapping from one topological space to another.
We discuss how this is a true generalization of the real valued continuous mappings discussed in analysis, and we give multiple alternative ways to describe continuity.
To solidify the concept we look at several examples of real mappings, some of which are continuous, and some of which are not.
Then, working towards the Weirstrass Intermediate Value Theorem, we show that every continuous surjection from one topological space to another preserves continuity. We also introduce the idea of path-connected spaces, which is an easy to visualize special type of continuity. We then formally define
the idea of intervals of real numbers, and show that a subset of reals is connected if and only if it is an interval. After this, we state and prove the Weirstrass Intermediate Value Theorem, and use it to establish the one dimensional version of the Brouwer Fixed Point Theorem (which states that a continuous mapping from the closed unit interval to itself must have a fixed point).
These lectures follow the book:
TopologyWithoutTears by SA Morrishttp://u.math.biu.ac.il/~megereli/topbook.pdf
My website is
https://sites.google.com/site/richardsouthwell254/home

The lecture was held within the framework of the Hausdorff Trimester Program : Applied and Computational Algebraic TopologyConcurrency theory in Computer Science studies effects that arise when several processors run simultaneously sharing common resources. It attempts to advise methods dealing with the "state space explosion problem" characterized by an exponentially growing number of execution paths; sometimes using models with a combinatorial/topological flavor. It is a common feature of these models that an execution corresponds to a directed path (d-path) in a (time-flow directed) state space, and that d-homotopies (preserving the directions) have equivalent computations as a result.
Getting to grips with the effects of the non-reversible time-flow is essential, and one needs to "twist" methods from ordinary algebraic topology in order to make them applicable. An essential task consists in inferring information about spaces of executions (directed paths) between two given states from information about the state space. The determination of path components is particularly important for applications.
I will discuss particular directed spaces arising from Higher Dimensional Automata (HDA). There are various methods identifying the homotopy type of the space of executions between two states in such an automaton with some finite complex: in simple cases as prodsimplicial complex – with products of simplices as building blocks – or as a configuration space living in a product of simplices. In several interesting cases, it is possible to give an explicit description of the homotopy type of the Alexander dual of such a configuration space and hence of the stable homotopy type of the corresponding trace space. This opens up for calculations of homology groups and of other topological invariants of some execution spaces.
We sketch a method recently devised by Ziemiański identifying – for a general HDA – a space of directed paths with a prodpermutahedral complex arising by glueing various permutahedra along their boundaries.
Joint work with L. Fajstrup (Aalborg), E. Goubault, E. Haucourt, S. Mimram (Éc. Polytechnique, Paris), R. Meshulam (Haifa) and K. Ziemiański (Warsaw).

Verifying shortest paths in a six-node topology on GENI

6:20

02 SSL Insight - Introduction to Basic SSLi Topologies

02 SSL Insight - Introduction to Basic SSLi Topologies

02 SSL Insight - Introduction to Basic SSLi Topologies

In this video, we can see the basic network topologies that an SSLi device can be configured into. The basic topologies that we look at here are the L2 SinglePath, L2 Multiple Path, L3 Single Path and L3 Multiple Path Topologies.

Path Deform + Topology

Mod-02 Lec-08 Homotopy and the First Fundamental Group

An Introduction toRiemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
- To understand the notion of homotopy of paths in a topological space
- To understand concatenation of paths in a topological space
- To sketch how the set of fixed-end-point (FEP) homotopy classes of loops at a point
becomes a group under concatenation, called the First FundamentalGroup
- To look at examples of fundamental groups of some common topological spaces
- To realise that the fundamental group is an algebraic invariant of topological spaces which
helps in distinguishing non-isomorphic topological spaces
- To...

AlgTop24: The fundamental group

This lecture introduces the fundamental group of a surface. We begin by discussing when two paths on a surface are homotopic, then defining multiplication of paths, and then multiplication of equivalence classes or types of loops based at a fixed point of the surface.
The fundamental groups of the disk and circle are described.
This is part of a beginner's course on Algebraic Topology given by N J Wildberger at UNSW.
A screenshot PDF which includes AlgTop21 to 29 can be found at my WildEgg website here: http://www.wildegg.com/store/p119/product-Algebraic-Topology-C-screenshot-pdf

I will explain a viewpoint on resurgence via Floer-type topology of Lefschetz thimbles in the complexified path integral. In particular, so called 2d-4d wall-crossing formalism of Gaiotto, Moore and Neitzke arises naturally in this framework, without any reference to supersymmetric field theories.

Jose Bravo demonstrates how to navigate through the QRM topology. Next he demonstrates the Configuration Monitor's ability to display the configurations of the network firewalls and routers in a vendor neutral way and to maintain a historical record of the configuration changes. He demonstrates how to highlight the differences between two sets of firewall configurations. He also demonstrate's QRM's ability to compare configurations between two different firewalls or any two devices.
Next, Jose demonstrates how QRM discovers new devices in the environment and he demonstrates how to search for devices in the topology so that missing machines can be located.
Jose also demonstrates QRM's capability to identify the path routes between two different machines and how the path between the two...

Recording of JonathanWilliamson's talk at the 2012BlenderConference on how to work with complex topology on organic and hard-surface models.
The video recording is pretty rough, the lighting in the room made it very difficult to get good footage. Thanks to the kind user that filmed the talk for us!

published: 29 Oct 2012

Network Theory Topology

Follow along with the course eBook: https://goo.gl/hhChUL
See the full course: http://complexitylabs.io/courses
We call the overall structure to a network its topology where topology simply means the way in which constituent parts are interrelated or arranged. To illustrate how network topology affects the system we look at a number of simple networks each containing the same amount of nodes but each having a different overall topology owing to the way they were connected with these topologies affecting different features to the network, such as the shortest path length or how we might control the flow of information on the network.
Twitter: https://goo.gl/Nu6Qap
Facebook: https://goo.gl/ggxGMT
LinkedIn:https://goo.gl/3v1vwF
Transcription Except:
In the previous module, we were discussin...

published: 21 Apr 2015

Graph Topological Sort Using Depth-First Search

In this video tutorial, you will learn how to do a topological sort on a directed acyclic graph (DAG), i.e. arrange vertices in a sequence according to dependency constraints shown by edges. The sort is done by using the depth-first search algorithm as the basis, and modifying it to label vertices with sequence numbers. The complete code is available at https://www.dropbox.com/s/71fe4ttpj27jxa8/Graph.java?dl=0
and the examples graphs illustrated in the video are at https://www.dropbox.com/s/6s8i9ji1wla28we/topsort_graph1.txt?dl=0
and
https://www.dropbox.com/s/tasix4uihrs1n5p/topsort_graph2.txt?dl=0
Error: At around 6:24, the graph example used has a cycle including X, C, D, P. The edge from X to C should go in the opposite direction, from C to X. Everything else works as it should.

published: 04 Aug 2016

ArcMap 45 - عملية تصحيح الاخطاء الهندسية Topology

published: 17 Mar 2017

Coverings of the Circle

A covering of a topological space X is a topological space Y together with a continuous surjective map from X to Y that is locally bi-continuos.
The infinite spiral is for example a covering of the circle. Notice how every path on the circle can be lifted to the spiral.
If a covering has a trivial fundamental group, i.e. it does not admit any non trivial closed paths it is called the universal covering. Here we see how a closed path on the circle is lifed to a non closed path on the spiral. Indeed the infinite spiral is the universal covering of the circle.
The name universal comes from an other property: The universal covering is a covering of every other covering. This is shown here with the 2:1 and the 3:1 covering of the circle. The universal covering covers both of them, b...

published: 08 Nov 2006

Undergraduate Topology: Feb 7, connectedness (part 1)

In this lecture we formalize the idea of a continuous mapping from one topological space to another.
We discuss how this is a true generalization of the real valued continuous mappings discussed in analysis, and we give multiple alternative ways to describe continuity.
To solidify the concept we look at several examples of real mappings, some of which are continuous, and some of which are not.
Then, working towards the Weirstrass Intermediate Value Theorem, we show that every continuous surjection from one topological space to another preserves continuity. We also introduce the idea of path-connected spaces, which is an easy to visualize special type of continuity. We then formally define
the idea of intervals of real numbers, and show that a subset of reals is connected if and only i...

The lecture was held within the framework of the Hausdorff Trimester Program : Applied and Computational Algebraic TopologyConcurrency theory in Computer Science studies effects that arise when several processors run simultaneously sharing common resources. It attempts to advise methods dealing with the "state space explosion problem" characterized by an exponentially growing number of execution paths; sometimes using models with a combinatorial/topological flavor. It is a common feature of these models that an execution corresponds to a directed path (d-path) in a (time-flow directed) state space, and that d-homotopies (preserving the directions) have equivalent computations as a result.
Getting to grips with the effects of the non-reversible time-flow is essential, and one needs to "t...

Path Deform + Topology

Verifying shortest paths in a six-node topology on GENI

published: 28 Mar 2017

02 SSL Insight - Introduction to Basic SSLi Topologies

In this video, we can see the basic network topologies that an SSLi device can be configured into. The basic topologies that we look at here are the L2 SinglePath, L2 Multiple Path, L3 Single Path and L3 Multiple Path Topologies.

An Introduction toRiemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
- To understand the notion of homotopy of paths in a topological space
- To understand concatenation of paths in a topological space
- To sketch how the set of fixed-end-point (FEP) homotopy classes of loops at a point
becomes a group under concatenation, called the First FundamentalGroup
- To look at examples of fundamental groups of some common topological spaces
- To realise that the fundamental group is an algebraic invariant of topological spaces which
helps in distinguishing non-isomorphic topological spaces
- To realise that a first classification of Riemann surfaces can be done based on their fundamental
groups by appealing to the theory of covering spaces
Keywords:
Path or arc in a topological space, initial or starting point and terminal or ending point of a path,
path as a map, geometric path, parametrisation of a geometric path,
homotopy, continuous deformation of maps, product topology, equivalence of
paths under homotopy, fixed-end-point (FEP) homotopy, concatenation of paths,
constant path, binary operation, associative binary operation, identity element for
a binary operation, inverse of an element under a binary operation, first fundamental group, topological invariant

An Introduction toRiemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
- To understand the notion of homotopy of paths in a topological space
- To understand concatenation of paths in a topological space
- To sketch how the set of fixed-end-point (FEP) homotopy classes of loops at a point
becomes a group under concatenation, called the First FundamentalGroup
- To look at examples of fundamental groups of some common topological spaces
- To realise that the fundamental group is an algebraic invariant of topological spaces which
helps in distinguishing non-isomorphic topological spaces
- To realise that a first classification of Riemann surfaces can be done based on their fundamental
groups by appealing to the theory of covering spaces
Keywords:
Path or arc in a topological space, initial or starting point and terminal or ending point of a path,
path as a map, geometric path, parametrisation of a geometric path,
homotopy, continuous deformation of maps, product topology, equivalence of
paths under homotopy, fixed-end-point (FEP) homotopy, concatenation of paths,
constant path, binary operation, associative binary operation, identity element for
a binary operation, inverse of an element under a binary operation, first fundamental group, topological invariant

AlgTop24: The fundamental group

This lecture introduces the fundamental group of a surface. We begin by discussing when two paths on a surface are homotopic, then defining multiplication of pa...

This lecture introduces the fundamental group of a surface. We begin by discussing when two paths on a surface are homotopic, then defining multiplication of paths, and then multiplication of equivalence classes or types of loops based at a fixed point of the surface.
The fundamental groups of the disk and circle are described.
This is part of a beginner's course on Algebraic Topology given by N J Wildberger at UNSW.
A screenshot PDF which includes AlgTop21 to 29 can be found at my WildEgg website here: http://www.wildegg.com/store/p119/product-Algebraic-Topology-C-screenshot-pdf

This lecture introduces the fundamental group of a surface. We begin by discussing when two paths on a surface are homotopic, then defining multiplication of paths, and then multiplication of equivalence classes or types of loops based at a fixed point of the surface.
The fundamental groups of the disk and circle are described.
This is part of a beginner's course on Algebraic Topology given by N J Wildberger at UNSW.
A screenshot PDF which includes AlgTop21 to 29 can be found at my WildEgg website here: http://www.wildegg.com/store/p119/product-Algebraic-Topology-C-screenshot-pdf

Who cares about topology? (Inscribed rectangle problem)

An unsolved conjecture, the inscribed square problem, and a clever topological solution to a weaker version of the question, the inscribed rectangle problem (Pr...

An unsolved conjecture, the inscribed square problem, and a clever topological solution to a weaker version of the question, the inscribed rectangle problem (Proof due to H. Vaughan, 1977), that shows how the torus and mobius strip naturally arise in mathematical ponderings.
Patreon: https://www.patreon.com/3blue1brown
Get 10% your domain name purchace from https://www.hover.com/, by using the promo code TOPOLOGY.
Special shout out to the following patrons: Dave Nicponski, Juan Batiz-Benet, Loo Yu Jun, Tom, Othman Alikhan, Markus Persson, Joseph John Cox, Achille Brighton, Kirk Werklund, Luc Ritchie, RiptaPasay, PatrickJMT , Felipe Diniz, Chris, Andrew Mcnab, Matt Parlmer, NaokiOrai, Dan Davison, JoseOscar Mur-Miranda, Aidan Boneham, BrentKennedy, Henry Reich, Sean Bibby, PaulConstantine, Justin Clark, Mohannad Elhamod, Denis, Ben Granger, Ali Yahya, Jeffrey Herman, and Jacob Young
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC-
Various social media stuffs:
Twitter: https://twitter.com/3Blue1Brown
Facebook: https://www.facebook.com/3blue1brown/
Reddit: https://www.reddit.com/r/3Blue1Brown

An unsolved conjecture, the inscribed square problem, and a clever topological solution to a weaker version of the question, the inscribed rectangle problem (Proof due to H. Vaughan, 1977), that shows how the torus and mobius strip naturally arise in mathematical ponderings.
Patreon: https://www.patreon.com/3blue1brown
Get 10% your domain name purchace from https://www.hover.com/, by using the promo code TOPOLOGY.
Special shout out to the following patrons: Dave Nicponski, Juan Batiz-Benet, Loo Yu Jun, Tom, Othman Alikhan, Markus Persson, Joseph John Cox, Achille Brighton, Kirk Werklund, Luc Ritchie, RiptaPasay, PatrickJMT , Felipe Diniz, Chris, Andrew Mcnab, Matt Parlmer, NaokiOrai, Dan Davison, JoseOscar Mur-Miranda, Aidan Boneham, BrentKennedy, Henry Reich, Sean Bibby, PaulConstantine, Justin Clark, Mohannad Elhamod, Denis, Ben Granger, Ali Yahya, Jeffrey Herman, and Jacob Young
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC-
Various social media stuffs:
Twitter: https://twitter.com/3Blue1Brown
Facebook: https://www.facebook.com/3blue1brown/
Reddit: https://www.reddit.com/r/3Blue1Brown

I will explain a viewpoint on resurgence via Floer-type topology of Lefschetz thimbles in the complexified path integral. In particular, so called 2d-4d wall-cr...

I will explain a viewpoint on resurgence via Floer-type topology of Lefschetz thimbles in the complexified path integral. In particular, so called 2d-4d wall-crossing formalism of Gaiotto, Moore and Neitzke arises naturally in this framework, without any reference to supersymmetric field theories.

I will explain a viewpoint on resurgence via Floer-type topology of Lefschetz thimbles in the complexified path integral. In particular, so called 2d-4d wall-crossing formalism of Gaiotto, Moore and Neitzke arises naturally in this framework, without any reference to supersymmetric field theories.

Jose Bravo demonstrates how to navigate through the QRM topology. Next he demonstrates the Configuration Monitor's ability to display the configurations of the ...

Jose Bravo demonstrates how to navigate through the QRM topology. Next he demonstrates the Configuration Monitor's ability to display the configurations of the network firewalls and routers in a vendor neutral way and to maintain a historical record of the configuration changes. He demonstrates how to highlight the differences between two sets of firewall configurations. He also demonstrate's QRM's ability to compare configurations between two different firewalls or any two devices.
Next, Jose demonstrates how QRM discovers new devices in the environment and he demonstrates how to search for devices in the topology so that missing machines can be located.
Jose also demonstrates QRM's capability to identify the path routes between two different machines and how the path between the two machines may be blocked.
IBM Owner: CalvinPowers

Jose Bravo demonstrates how to navigate through the QRM topology. Next he demonstrates the Configuration Monitor's ability to display the configurations of the network firewalls and routers in a vendor neutral way and to maintain a historical record of the configuration changes. He demonstrates how to highlight the differences between two sets of firewall configurations. He also demonstrate's QRM's ability to compare configurations between two different firewalls or any two devices.
Next, Jose demonstrates how QRM discovers new devices in the environment and he demonstrates how to search for devices in the topology so that missing machines can be located.
Jose also demonstrates QRM's capability to identify the path routes between two different machines and how the path between the two machines may be blocked.
IBM Owner: CalvinPowers

Recording of JonathanWilliamson's talk at the 2012BlenderConference on how to work with complex topology on organic and hard-surface models.
The video recording is pretty rough, the lighting in the room made it very difficult to get good footage. Thanks to the kind user that filmed the talk for us!

Recording of JonathanWilliamson's talk at the 2012BlenderConference on how to work with complex topology on organic and hard-surface models.
The video recording is pretty rough, the lighting in the room made it very difficult to get good footage. Thanks to the kind user that filmed the talk for us!

Network Theory Topology

Follow along with the course eBook: https://goo.gl/hhChUL
See the full course: http://complexitylabs.io/courses
We call the overall structure to a network its t...

Follow along with the course eBook: https://goo.gl/hhChUL
See the full course: http://complexitylabs.io/courses
We call the overall structure to a network its topology where topology simply means the way in which constituent parts are interrelated or arranged. To illustrate how network topology affects the system we look at a number of simple networks each containing the same amount of nodes but each having a different overall topology owing to the way they were connected with these topologies affecting different features to the network, such as the shortest path length or how we might control the flow of information on the network.
Twitter: https://goo.gl/Nu6Qap
Facebook: https://goo.gl/ggxGMT
LinkedIn:https://goo.gl/3v1vwF
Transcription Except:
In the previous module, we were discussing local network metrics that referred to a single node in a graph, asking how connected it was or how influential and central it was while also building up our basic vocabulary from graph theory. In the coming set of lectures we are going to be looking at global metrics that refer to the whole of graph. A we have previously noted networks are a very informal type of structure they often simple develop without any overall top down design. Someone builds a protocol for two computers to exchange information over a network and shares it with a colleague, other people see the utility of it and connects to this little networks and then more people as the network grow, until 25 years later we have a massive network of networks that is the internet.
No one planned the internet just as one really planed the global financial networks that have emerged over the past few decades, traders, investor and institutions set up connections wherever they thought there was a viable return on investment, but now that these networks are here there overall markup feeds back to effect us the user, networks may start out quite random but they often develop into some stable overall structure and understanding the patterns to this overall structure is of central importance in network theory and the focus of this section to the course.
We can call this overall structure to a network its topology, where topology simply means the way in which constituent parts are interrelated or arranged:
Within the context of a network it defines the way different nodes are placed and interconnected with each other and the overall patterns that emerge out of this.
To illustrate this further here are a set of simple networks each containing the same amount of nodes but each having a different overall topology owing to the way it is connected. We can see how these different network topologies would in turn have very different features and properties to them. Imagine they where different transportation systems in which you are trying to get form A to B. In the star topology it would only ever take you two hops to reach your destination but in the ring it might take 3, the tree structure possibly 4, and this same influence from the topology would apply if we were trying to rout water through a hydraulic network or electricity on a power grid.
This network could also represent the flow of information within a national society we can see how the centralized star structure or the tree structure would be much easier for some political regime to control and influence as opposed to the more decentralized fully connected or ring network. The point I are trying to make clear is simply that networks have overall structure and this overall topology to the network matters as it feedback to affect the actions and capabilities of the nodes on the local level.

Follow along with the course eBook: https://goo.gl/hhChUL
See the full course: http://complexitylabs.io/courses
We call the overall structure to a network its topology where topology simply means the way in which constituent parts are interrelated or arranged. To illustrate how network topology affects the system we look at a number of simple networks each containing the same amount of nodes but each having a different overall topology owing to the way they were connected with these topologies affecting different features to the network, such as the shortest path length or how we might control the flow of information on the network.
Twitter: https://goo.gl/Nu6Qap
Facebook: https://goo.gl/ggxGMT
LinkedIn:https://goo.gl/3v1vwF
Transcription Except:
In the previous module, we were discussing local network metrics that referred to a single node in a graph, asking how connected it was or how influential and central it was while also building up our basic vocabulary from graph theory. In the coming set of lectures we are going to be looking at global metrics that refer to the whole of graph. A we have previously noted networks are a very informal type of structure they often simple develop without any overall top down design. Someone builds a protocol for two computers to exchange information over a network and shares it with a colleague, other people see the utility of it and connects to this little networks and then more people as the network grow, until 25 years later we have a massive network of networks that is the internet.
No one planned the internet just as one really planed the global financial networks that have emerged over the past few decades, traders, investor and institutions set up connections wherever they thought there was a viable return on investment, but now that these networks are here there overall markup feeds back to effect us the user, networks may start out quite random but they often develop into some stable overall structure and understanding the patterns to this overall structure is of central importance in network theory and the focus of this section to the course.
We can call this overall structure to a network its topology, where topology simply means the way in which constituent parts are interrelated or arranged:
Within the context of a network it defines the way different nodes are placed and interconnected with each other and the overall patterns that emerge out of this.
To illustrate this further here are a set of simple networks each containing the same amount of nodes but each having a different overall topology owing to the way it is connected. We can see how these different network topologies would in turn have very different features and properties to them. Imagine they where different transportation systems in which you are trying to get form A to B. In the star topology it would only ever take you two hops to reach your destination but in the ring it might take 3, the tree structure possibly 4, and this same influence from the topology would apply if we were trying to rout water through a hydraulic network or electricity on a power grid.
This network could also represent the flow of information within a national society we can see how the centralized star structure or the tree structure would be much easier for some political regime to control and influence as opposed to the more decentralized fully connected or ring network. The point I are trying to make clear is simply that networks have overall structure and this overall topology to the network matters as it feedback to affect the actions and capabilities of the nodes on the local level.

Graph Topological Sort Using Depth-First Search

In this video tutorial, you will learn how to do a topological sort on a directed acyclic graph (DAG), i.e. arrange vertices in a sequence according to dependen...

In this video tutorial, you will learn how to do a topological sort on a directed acyclic graph (DAG), i.e. arrange vertices in a sequence according to dependency constraints shown by edges. The sort is done by using the depth-first search algorithm as the basis, and modifying it to label vertices with sequence numbers. The complete code is available at https://www.dropbox.com/s/71fe4ttpj27jxa8/Graph.java?dl=0
and the examples graphs illustrated in the video are at https://www.dropbox.com/s/6s8i9ji1wla28we/topsort_graph1.txt?dl=0
and
https://www.dropbox.com/s/tasix4uihrs1n5p/topsort_graph2.txt?dl=0
Error: At around 6:24, the graph example used has a cycle including X, C, D, P. The edge from X to C should go in the opposite direction, from C to X. Everything else works as it should.

In this video tutorial, you will learn how to do a topological sort on a directed acyclic graph (DAG), i.e. arrange vertices in a sequence according to dependency constraints shown by edges. The sort is done by using the depth-first search algorithm as the basis, and modifying it to label vertices with sequence numbers. The complete code is available at https://www.dropbox.com/s/71fe4ttpj27jxa8/Graph.java?dl=0
and the examples graphs illustrated in the video are at https://www.dropbox.com/s/6s8i9ji1wla28we/topsort_graph1.txt?dl=0
and
https://www.dropbox.com/s/tasix4uihrs1n5p/topsort_graph2.txt?dl=0
Error: At around 6:24, the graph example used has a cycle including X, C, D, P. The edge from X to C should go in the opposite direction, from C to X. Everything else works as it should.

Coverings of the Circle

A covering of a topological space X is a topological space Y together with a continuous surjective map from X to Y that is locally bi-continuos.
The infinit...

A covering of a topological space X is a topological space Y together with a continuous surjective map from X to Y that is locally bi-continuos.
The infinite spiral is for example a covering of the circle. Notice how every path on the circle can be lifted to the spiral.
If a covering has a trivial fundamental group, i.e. it does not admit any non trivial closed paths it is called the universal covering. Here we see how a closed path on the circle is lifed to a non closed path on the spiral. Indeed the infinite spiral is the universal covering of the circle.
The name universal comes from an other property: The universal covering is a covering of every other covering. This is shown here with the 2:1 and the 3:1 covering of the circle. The universal covering covers both of them, but the 3:1 does not cover the 2:1.
From this universality property it follows also that every topological space has a unique universal covering. (not shown)
This Video was produces for a topology seminar at the LeibnizUniversitaet Hannover.
http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1

A covering of a topological space X is a topological space Y together with a continuous surjective map from X to Y that is locally bi-continuos.
The infinite spiral is for example a covering of the circle. Notice how every path on the circle can be lifted to the spiral.
If a covering has a trivial fundamental group, i.e. it does not admit any non trivial closed paths it is called the universal covering. Here we see how a closed path on the circle is lifed to a non closed path on the spiral. Indeed the infinite spiral is the universal covering of the circle.
The name universal comes from an other property: The universal covering is a covering of every other covering. This is shown here with the 2:1 and the 3:1 covering of the circle. The universal covering covers both of them, but the 3:1 does not cover the 2:1.
From this universality property it follows also that every topological space has a unique universal covering. (not shown)
This Video was produces for a topology seminar at the LeibnizUniversitaet Hannover.
http://www-ifm.math.uni-hannover.de/~fugru/?topologie_teil1

In this lecture we formalize the idea of a continuous mapping from one topological space to another.
We discuss how this is a true generalization of the real v...

In this lecture we formalize the idea of a continuous mapping from one topological space to another.
We discuss how this is a true generalization of the real valued continuous mappings discussed in analysis, and we give multiple alternative ways to describe continuity.
To solidify the concept we look at several examples of real mappings, some of which are continuous, and some of which are not.
Then, working towards the Weirstrass Intermediate Value Theorem, we show that every continuous surjection from one topological space to another preserves continuity. We also introduce the idea of path-connected spaces, which is an easy to visualize special type of continuity. We then formally define
the idea of intervals of real numbers, and show that a subset of reals is connected if and only if it is an interval. After this, we state and prove the Weirstrass Intermediate Value Theorem, and use it to establish the one dimensional version of the Brouwer Fixed Point Theorem (which states that a continuous mapping from the closed unit interval to itself must have a fixed point).
These lectures follow the book:
TopologyWithoutTears by SA Morrishttp://u.math.biu.ac.il/~megereli/topbook.pdf
My website is
https://sites.google.com/site/richardsouthwell254/home

In this lecture we formalize the idea of a continuous mapping from one topological space to another.
We discuss how this is a true generalization of the real valued continuous mappings discussed in analysis, and we give multiple alternative ways to describe continuity.
To solidify the concept we look at several examples of real mappings, some of which are continuous, and some of which are not.
Then, working towards the Weirstrass Intermediate Value Theorem, we show that every continuous surjection from one topological space to another preserves continuity. We also introduce the idea of path-connected spaces, which is an easy to visualize special type of continuity. We then formally define
the idea of intervals of real numbers, and show that a subset of reals is connected if and only if it is an interval. After this, we state and prove the Weirstrass Intermediate Value Theorem, and use it to establish the one dimensional version of the Brouwer Fixed Point Theorem (which states that a continuous mapping from the closed unit interval to itself must have a fixed point).
These lectures follow the book:
TopologyWithoutTears by SA Morrishttp://u.math.biu.ac.il/~megereli/topbook.pdf
My website is
https://sites.google.com/site/richardsouthwell254/home

The lecture was held within the framework of the Hausdorff Trimester Program : Applied and Computational Algebraic TopologyConcurrency theory in Computer Science studies effects that arise when several processors run simultaneously sharing common resources. It attempts to advise methods dealing with the "state space explosion problem" characterized by an exponentially growing number of execution paths; sometimes using models with a combinatorial/topological flavor. It is a common feature of these models that an execution corresponds to a directed path (d-path) in a (time-flow directed) state space, and that d-homotopies (preserving the directions) have equivalent computations as a result.
Getting to grips with the effects of the non-reversible time-flow is essential, and one needs to "twist" methods from ordinary algebraic topology in order to make them applicable. An essential task consists in inferring information about spaces of executions (directed paths) between two given states from information about the state space. The determination of path components is particularly important for applications.
I will discuss particular directed spaces arising from Higher Dimensional Automata (HDA). There are various methods identifying the homotopy type of the space of executions between two states in such an automaton with some finite complex: in simple cases as prodsimplicial complex – with products of simplices as building blocks – or as a configuration space living in a product of simplices. In several interesting cases, it is possible to give an explicit description of the homotopy type of the Alexander dual of such a configuration space and hence of the stable homotopy type of the corresponding trace space. This opens up for calculations of homology groups and of other topological invariants of some execution spaces.
We sketch a method recently devised by Ziemiański identifying – for a general HDA – a space of directed paths with a prodpermutahedral complex arising by glueing various permutahedra along their boundaries.
Joint work with L. Fajstrup (Aalborg), E. Goubault, E. Haucourt, S. Mimram (Éc. Polytechnique, Paris), R. Meshulam (Haifa) and K. Ziemiański (Warsaw).

The lecture was held within the framework of the Hausdorff Trimester Program : Applied and Computational Algebraic TopologyConcurrency theory in Computer Science studies effects that arise when several processors run simultaneously sharing common resources. It attempts to advise methods dealing with the "state space explosion problem" characterized by an exponentially growing number of execution paths; sometimes using models with a combinatorial/topological flavor. It is a common feature of these models that an execution corresponds to a directed path (d-path) in a (time-flow directed) state space, and that d-homotopies (preserving the directions) have equivalent computations as a result.
Getting to grips with the effects of the non-reversible time-flow is essential, and one needs to "twist" methods from ordinary algebraic topology in order to make them applicable. An essential task consists in inferring information about spaces of executions (directed paths) between two given states from information about the state space. The determination of path components is particularly important for applications.
I will discuss particular directed spaces arising from Higher Dimensional Automata (HDA). There are various methods identifying the homotopy type of the space of executions between two states in such an automaton with some finite complex: in simple cases as prodsimplicial complex – with products of simplices as building blocks – or as a configuration space living in a product of simplices. In several interesting cases, it is possible to give an explicit description of the homotopy type of the Alexander dual of such a configuration space and hence of the stable homotopy type of the corresponding trace space. This opens up for calculations of homology groups and of other topological invariants of some execution spaces.
We sketch a method recently devised by Ziemiański identifying – for a general HDA – a space of directed paths with a prodpermutahedral complex arising by glueing various permutahedra along their boundaries.
Joint work with L. Fajstrup (Aalborg), E. Goubault, E. Haucourt, S. Mimram (Éc. Polytechnique, Paris), R. Meshulam (Haifa) and K. Ziemiański (Warsaw).

02 SSL Insight - Introduction to Basic SSLi Topologies

In this video, we can see the basic network topologies that an SSLi device can be configured into. The basic topologies that we look at here are the L2 Single P...

In this video, we can see the basic network topologies that an SSLi device can be configured into. The basic topologies that we look at here are the L2 SinglePath, L2 Multiple Path, L3 Single Path and L3 Multiple Path Topologies.

In this video, we can see the basic network topologies that an SSLi device can be configured into. The basic topologies that we look at here are the L2 SinglePath, L2 Multiple Path, L3 Single Path and L3 Multiple Path Topologies.

Mod-02 Lec-08 Homotopy and the First Fundamental Group

An Introduction toRiemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
- To understand the notion of homotopy of paths in a topological space
- To understand concatenation of paths in a topological space
- To sketch how the set of fixed-end-point (FEP) homotopy classes of loops at a point
becomes a group under concatenation, called the First FundamentalGroup
- To look at examples of fundamental groups of some common topological spaces
- To realise that the fundamental group is an algebraic invariant of topological spaces which
helps in distinguishing non-isomorphic topological spaces
- To...

published: 14 Jun 2013

AlgTop24: The fundamental group

This lecture introduces the fundamental group of a surface. We begin by discussing when two paths on a surface are homotopic, then defining multiplication of paths, and then multiplication of equivalence classes or types of loops based at a fixed point of the surface.
The fundamental groups of the disk and circle are described.
This is part of a beginner's course on Algebraic Topology given by N J Wildberger at UNSW.
A screenshot PDF which includes AlgTop21 to 29 can be found at my WildEgg website here: http://www.wildegg.com/store/p119/product-Algebraic-Topology-C-screenshot-pdf

I will explain a viewpoint on resurgence via Floer-type topology of Lefschetz thimbles in the complexified path integral. In particular, so called 2d-4d wall-crossing formalism of Gaiotto, Moore and Neitzke arises naturally in this framework, without any reference to supersymmetric field theories.

The lecture was held within the framework of the Hausdorff Trimester Program : Applied and Computational Algebraic TopologyConcurrency theory in Computer Science studies effects that arise when several processors run simultaneously sharing common resources. It attempts to advise methods dealing with the "state space explosion problem" characterized by an exponentially growing number of execution paths; sometimes using models with a combinatorial/topological flavor. It is a common feature of these models that an execution corresponds to a directed path (d-path) in a (time-flow directed) state space, and that d-homotopies (preserving the directions) have equivalent computations as a result.
Getting to grips with the effects of the non-reversible time-flow is essential, and one needs to "t...

Configuration Spaces and Topology of Robot Motion Planning

AlgTop28: Covering spaces and fundamental groups

We illustrate the ideas from the last lectures by giving some more examples of covering spaces: of the torus, and the two-holed torus. Then we begin to explore the relationship between the fundamental groups of a covering space X and a base space B under a covering map p:X to B.
For this we need two important Lemmas: the Lifting Path lemma, and the Lifting Homotopy lemma. Then we obtain the basic result that the fundamental group of X can be viewed as a subgroup of the fundamental group of the base B, via the induced homomorphism of p. So the possibility emerges of studying covering spaces of B by studying subgroups of the fundamental group of B.
A screenshot PDF which includes AlgTop21 to 29 can be found at my WildEgg website here: http://www.wildegg.com/store/p119/product-Algebraic-Top...

EE 101/10 - Topology, nodes, branches, series, parallel connections.

Basic as it sounds, the definitions in this short video are the linchpin of everything that follows. The bullet proof definitions will provide us with the necessary platform to analyze the most complicated electronic circuits that technology may throw at us.

published: 01 Aug 2015

What is a Manifold? Lesson 5: Compactness, Connectedness, and Topological Properties

The last lesson covering the topological prep-work required before we begin the discussion of manifolds. Topics covered: compactness, connectedness, and the relationship between homeomorphisms and topological properties.

P. Friz - Rough paths, IHP 06/10/2014 - Part 1

Using Blender's Light Path Node for Cycles

Learn how to use the LightPath node for Cycleshttp://cgcookie.com/blender/2013/02/26/blender-cycles-light-path-node/
Blender's light path node is incredibly useful for all sorts of tasks, but it's also a bit difficult to grasp at times and so many of us are left with no understanding of how to use it. This tutorial gives you an introduction to the light path node, showing you what can be done with it in Cycles. However, it's easier to understand how to use this tool when we know how render engines work, so we will give you a basic introduction to raytracing. You'll also learn why it works, and what you need to know in order to use it effectively.
We will be taking a look at how to control the influence of created materials on other objects in our scene. This includes changing the colo...

published: 26 Feb 2013

Introduction to the fundamental group

A leisurely introduction to the idea of homotopy, including a sketch proof that the fundamental group of the circle is the integers. This is aimed at second year undergrads embarking on projects in geometry/topology who need to quickly learn the basic idea of the fundamental group, so the focus is on developing intuition and explaining the definitions.

In this lecture we formalize the idea of a continuous mapping from one topological space to another.
We discuss how this is a true generalization of the real valued continuous mappings discussed in analysis, and we give multiple alternative ways to describe continuity.
To solidify the concept we look at several examples of real mappings, some of which are continuous, and some of which are not.
Then, working towards the Weirstrass Intermediate Value Theorem, we show that every continuous surjection from one topological space to another preserves continuity. We also introduce the idea of path-connected spaces, which is an easy to visualize special type of continuity. We then formally define
the idea of intervals of real numbers, and show that a subset of reals is connected if and only i...

Undergraduate Topology: Feb 7, connectedness (part 1)

Recording of JonathanWilliamson's talk at the 2012BlenderConference on how to work with complex topology on organic and hard-surface models.
The video recording is pretty rough, the lighting in the room made it very difficult to get good footage. Thanks to the kind user that filmed the talk for us!

An Introduction toRiemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
- To understand the notion of homotopy of paths in a topological space
- To understand concatenation of paths in a topological space
- To sketch how the set of fixed-end-point (FEP) homotopy classes of loops at a point
becomes a group under concatenation, called the First FundamentalGroup
- To look at examples of fundamental groups of some common topological spaces
- To realise that the fundamental group is an algebraic invariant of topological spaces which
helps in distinguishing non-isomorphic topological spaces
- To realise that a first classification of Riemann surfaces can be done based on their fundamental
groups by appealing to the theory of covering spaces
Keywords:
Path or arc in a topological space, initial or starting point and terminal or ending point of a path,
path as a map, geometric path, parametrisation of a geometric path,
homotopy, continuous deformation of maps, product topology, equivalence of
paths under homotopy, fixed-end-point (FEP) homotopy, concatenation of paths,
constant path, binary operation, associative binary operation, identity element for
a binary operation, inverse of an element under a binary operation, first fundamental group, topological invariant

An Introduction toRiemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
- To understand the notion of homotopy of paths in a topological space
- To understand concatenation of paths in a topological space
- To sketch how the set of fixed-end-point (FEP) homotopy classes of loops at a point
becomes a group under concatenation, called the First FundamentalGroup
- To look at examples of fundamental groups of some common topological spaces
- To realise that the fundamental group is an algebraic invariant of topological spaces which
helps in distinguishing non-isomorphic topological spaces
- To realise that a first classification of Riemann surfaces can be done based on their fundamental
groups by appealing to the theory of covering spaces
Keywords:
Path or arc in a topological space, initial or starting point and terminal or ending point of a path,
path as a map, geometric path, parametrisation of a geometric path,
homotopy, continuous deformation of maps, product topology, equivalence of
paths under homotopy, fixed-end-point (FEP) homotopy, concatenation of paths,
constant path, binary operation, associative binary operation, identity element for
a binary operation, inverse of an element under a binary operation, first fundamental group, topological invariant

AlgTop24: The fundamental group

This lecture introduces the fundamental group of a surface. We begin by discussing when two paths on a surface are homotopic, then defining multiplication of pa...

This lecture introduces the fundamental group of a surface. We begin by discussing when two paths on a surface are homotopic, then defining multiplication of paths, and then multiplication of equivalence classes or types of loops based at a fixed point of the surface.
The fundamental groups of the disk and circle are described.
This is part of a beginner's course on Algebraic Topology given by N J Wildberger at UNSW.
A screenshot PDF which includes AlgTop21 to 29 can be found at my WildEgg website here: http://www.wildegg.com/store/p119/product-Algebraic-Topology-C-screenshot-pdf

This lecture introduces the fundamental group of a surface. We begin by discussing when two paths on a surface are homotopic, then defining multiplication of paths, and then multiplication of equivalence classes or types of loops based at a fixed point of the surface.
The fundamental groups of the disk and circle are described.
This is part of a beginner's course on Algebraic Topology given by N J Wildberger at UNSW.
A screenshot PDF which includes AlgTop21 to 29 can be found at my WildEgg website here: http://www.wildegg.com/store/p119/product-Algebraic-Topology-C-screenshot-pdf

I will explain a viewpoint on resurgence via Floer-type topology of Lefschetz thimbles in the complexified path integral. In particular, so called 2d-4d wall-cr...

I will explain a viewpoint on resurgence via Floer-type topology of Lefschetz thimbles in the complexified path integral. In particular, so called 2d-4d wall-crossing formalism of Gaiotto, Moore and Neitzke arises naturally in this framework, without any reference to supersymmetric field theories.

I will explain a viewpoint on resurgence via Floer-type topology of Lefschetz thimbles in the complexified path integral. In particular, so called 2d-4d wall-crossing formalism of Gaiotto, Moore and Neitzke arises naturally in this framework, without any reference to supersymmetric field theories.

The lecture was held within the framework of the Hausdorff Trimester Program : Applied and Computational Algebraic TopologyConcurrency theory in Computer Science studies effects that arise when several processors run simultaneously sharing common resources. It attempts to advise methods dealing with the "state space explosion problem" characterized by an exponentially growing number of execution paths; sometimes using models with a combinatorial/topological flavor. It is a common feature of these models that an execution corresponds to a directed path (d-path) in a (time-flow directed) state space, and that d-homotopies (preserving the directions) have equivalent computations as a result.
Getting to grips with the effects of the non-reversible time-flow is essential, and one needs to "twist" methods from ordinary algebraic topology in order to make them applicable. An essential task consists in inferring information about spaces of executions (directed paths) between two given states from information about the state space. The determination of path components is particularly important for applications.
I will discuss particular directed spaces arising from Higher Dimensional Automata (HDA). There are various methods identifying the homotopy type of the space of executions between two states in such an automaton with some finite complex: in simple cases as prodsimplicial complex – with products of simplices as building blocks – or as a configuration space living in a product of simplices. In several interesting cases, it is possible to give an explicit description of the homotopy type of the Alexander dual of such a configuration space and hence of the stable homotopy type of the corresponding trace space. This opens up for calculations of homology groups and of other topological invariants of some execution spaces.
We sketch a method recently devised by Ziemiański identifying – for a general HDA – a space of directed paths with a prodpermutahedral complex arising by glueing various permutahedra along their boundaries.
Joint work with L. Fajstrup (Aalborg), E. Goubault, E. Haucourt, S. Mimram (Éc. Polytechnique, Paris), R. Meshulam (Haifa) and K. Ziemiański (Warsaw).

The lecture was held within the framework of the Hausdorff Trimester Program : Applied and Computational Algebraic TopologyConcurrency theory in Computer Science studies effects that arise when several processors run simultaneously sharing common resources. It attempts to advise methods dealing with the "state space explosion problem" characterized by an exponentially growing number of execution paths; sometimes using models with a combinatorial/topological flavor. It is a common feature of these models that an execution corresponds to a directed path (d-path) in a (time-flow directed) state space, and that d-homotopies (preserving the directions) have equivalent computations as a result.
Getting to grips with the effects of the non-reversible time-flow is essential, and one needs to "twist" methods from ordinary algebraic topology in order to make them applicable. An essential task consists in inferring information about spaces of executions (directed paths) between two given states from information about the state space. The determination of path components is particularly important for applications.
I will discuss particular directed spaces arising from Higher Dimensional Automata (HDA). There are various methods identifying the homotopy type of the space of executions between two states in such an automaton with some finite complex: in simple cases as prodsimplicial complex – with products of simplices as building blocks – or as a configuration space living in a product of simplices. In several interesting cases, it is possible to give an explicit description of the homotopy type of the Alexander dual of such a configuration space and hence of the stable homotopy type of the corresponding trace space. This opens up for calculations of homology groups and of other topological invariants of some execution spaces.
We sketch a method recently devised by Ziemiański identifying – for a general HDA – a space of directed paths with a prodpermutahedral complex arising by glueing various permutahedra along their boundaries.
Joint work with L. Fajstrup (Aalborg), E. Goubault, E. Haucourt, S. Mimram (Éc. Polytechnique, Paris), R. Meshulam (Haifa) and K. Ziemiański (Warsaw).

AlgTop28: Covering spaces and fundamental groups

We illustrate the ideas from the last lectures by giving some more examples of covering spaces: of the torus, and the two-holed torus. Then we begin to explore ...

We illustrate the ideas from the last lectures by giving some more examples of covering spaces: of the torus, and the two-holed torus. Then we begin to explore the relationship between the fundamental groups of a covering space X and a base space B under a covering map p:X to B.
For this we need two important Lemmas: the Lifting Path lemma, and the Lifting Homotopy lemma. Then we obtain the basic result that the fundamental group of X can be viewed as a subgroup of the fundamental group of the base B, via the induced homomorphism of p. So the possibility emerges of studying covering spaces of B by studying subgroups of the fundamental group of B.
A screenshot PDF which includes AlgTop21 to 29 can be found at my WildEgg website here: http://www.wildegg.com/store/p119/product-Algebraic-Topology-C-screenshot-pdf

We illustrate the ideas from the last lectures by giving some more examples of covering spaces: of the torus, and the two-holed torus. Then we begin to explore the relationship between the fundamental groups of a covering space X and a base space B under a covering map p:X to B.
For this we need two important Lemmas: the Lifting Path lemma, and the Lifting Homotopy lemma. Then we obtain the basic result that the fundamental group of X can be viewed as a subgroup of the fundamental group of the base B, via the induced homomorphism of p. So the possibility emerges of studying covering spaces of B by studying subgroups of the fundamental group of B.
A screenshot PDF which includes AlgTop21 to 29 can be found at my WildEgg website here: http://www.wildegg.com/store/p119/product-Algebraic-Topology-C-screenshot-pdf

EE 101/10 - Topology, nodes, branches, series, parallel connections.

Basic as it sounds, the definitions in this short video are the linchpin of everything that follows. The bullet proof definitions will provide us with the neces...

Basic as it sounds, the definitions in this short video are the linchpin of everything that follows. The bullet proof definitions will provide us with the necessary platform to analyze the most complicated electronic circuits that technology may throw at us.

Basic as it sounds, the definitions in this short video are the linchpin of everything that follows. The bullet proof definitions will provide us with the necessary platform to analyze the most complicated electronic circuits that technology may throw at us.

published:01 Aug 2015

views:5629

back

What is a Manifold? Lesson 5: Compactness, Connectedness, and Topological Properties

The last lesson covering the topological prep-work required before we begin the discussion of manifolds. Topics covered: compactness, connectedness, and the rel...

The last lesson covering the topological prep-work required before we begin the discussion of manifolds. Topics covered: compactness, connectedness, and the relationship between homeomorphisms and topological properties.

The last lesson covering the topological prep-work required before we begin the discussion of manifolds. Topics covered: compactness, connectedness, and the relationship between homeomorphisms and topological properties.

Learn how to use the LightPath node for Cycleshttp://cgcookie.com/blender/2013/02/26/blender-cycles-light-path-node/
Blender's light path node is incredibly useful for all sorts of tasks, but it's also a bit difficult to grasp at times and so many of us are left with no understanding of how to use it. This tutorial gives you an introduction to the light path node, showing you what can be done with it in Cycles. However, it's easier to understand how to use this tool when we know how render engines work, so we will give you a basic introduction to raytracing. You'll also learn why it works, and what you need to know in order to use it effectively.
We will be taking a look at how to control the influence of created materials on other objects in our scene. This includes changing the color of an object inside a reflection, in the shadow, through refraction, or any number of other ways. It also covers how to hide objects from specific light paths, such that it appears invisible in reflections, doesn't cast shadows, etc.

Learn how to use the LightPath node for Cycleshttp://cgcookie.com/blender/2013/02/26/blender-cycles-light-path-node/
Blender's light path node is incredibly useful for all sorts of tasks, but it's also a bit difficult to grasp at times and so many of us are left with no understanding of how to use it. This tutorial gives you an introduction to the light path node, showing you what can be done with it in Cycles. However, it's easier to understand how to use this tool when we know how render engines work, so we will give you a basic introduction to raytracing. You'll also learn why it works, and what you need to know in order to use it effectively.
We will be taking a look at how to control the influence of created materials on other objects in our scene. This includes changing the color of an object inside a reflection, in the shadow, through refraction, or any number of other ways. It also covers how to hide objects from specific light paths, such that it appears invisible in reflections, doesn't cast shadows, etc.

Introduction to the fundamental group

A leisurely introduction to the idea of homotopy, including a sketch proof that the fundamental group of the circle is the integers. This is aimed at second yea...

A leisurely introduction to the idea of homotopy, including a sketch proof that the fundamental group of the circle is the integers. This is aimed at second year undergrads embarking on projects in geometry/topology who need to quickly learn the basic idea of the fundamental group, so the focus is on developing intuition and explaining the definitions.

A leisurely introduction to the idea of homotopy, including a sketch proof that the fundamental group of the circle is the integers. This is aimed at second year undergrads embarking on projects in geometry/topology who need to quickly learn the basic idea of the fundamental group, so the focus is on developing intuition and explaining the definitions.

In this lecture we formalize the idea of a continuous mapping from one topological space to another.
We discuss how this is a true generalization of the real v...

In this lecture we formalize the idea of a continuous mapping from one topological space to another.
We discuss how this is a true generalization of the real valued continuous mappings discussed in analysis, and we give multiple alternative ways to describe continuity.
To solidify the concept we look at several examples of real mappings, some of which are continuous, and some of which are not.
Then, working towards the Weirstrass Intermediate Value Theorem, we show that every continuous surjection from one topological space to another preserves continuity. We also introduce the idea of path-connected spaces, which is an easy to visualize special type of continuity. We then formally define
the idea of intervals of real numbers, and show that a subset of reals is connected if and only if it is an interval. After this, we state and prove the Weirstrass Intermediate Value Theorem, and use it to establish the one dimensional version of the Brouwer Fixed Point Theorem (which states that a continuous mapping from the closed unit interval to itself must have a fixed point).
These lectures follow the book:
TopologyWithoutTears by SA Morrishttp://u.math.biu.ac.il/~megereli/topbook.pdf
My website is
https://sites.google.com/site/richardsouthwell254/home

In this lecture we formalize the idea of a continuous mapping from one topological space to another.
We discuss how this is a true generalization of the real valued continuous mappings discussed in analysis, and we give multiple alternative ways to describe continuity.
To solidify the concept we look at several examples of real mappings, some of which are continuous, and some of which are not.
Then, working towards the Weirstrass Intermediate Value Theorem, we show that every continuous surjection from one topological space to another preserves continuity. We also introduce the idea of path-connected spaces, which is an easy to visualize special type of continuity. We then formally define
the idea of intervals of real numbers, and show that a subset of reals is connected if and only if it is an interval. After this, we state and prove the Weirstrass Intermediate Value Theorem, and use it to establish the one dimensional version of the Brouwer Fixed Point Theorem (which states that a continuous mapping from the closed unit interval to itself must have a fixed point).
These lectures follow the book:
TopologyWithoutTears by SA Morrishttp://u.math.biu.ac.il/~megereli/topbook.pdf
My website is
https://sites.google.com/site/richardsouthwell254/home

Recording of JonathanWilliamson's talk at the 2012BlenderConference on how to work with complex topology on organic and hard-surface models.
The video recording is pretty rough, the lighting in the room made it very difficult to get good footage. Thanks to the kind user that filmed the talk for us!

Recording of JonathanWilliamson's talk at the 2012BlenderConference on how to work with complex topology on organic and hard-surface models.
The video recording is pretty rough, the lighting in the room made it very difficult to get good footage. Thanks to the kind user that filmed the talk for us!

Mod-02 Lec-08 Homotopy and the First Fundamental Group

An Introduction toRiemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
- To understand the notion of homotopy of paths in a topological space
- To understand concatenation of paths in a topological space
- To sketch how the set of fixed-end-point (FEP) homotopy classes of loops at a point
becomes a group under concatenation, called the First FundamentalGroup
- To look at examples of fundamental groups of some common topological spaces
- To realise that the fundamental group is an algebraic invariant of topological spaces which
helps in distinguishing non-isomorphic topological spaces
- To realise that a first classification of Riemann surfaces can be done based on their fundamental
groups by appealing to the theory of covering spaces
Keywords:
Path or arc in a topological space, initial or starting point and terminal or ending point of a path,
path as a map, geometric path, parametrisation of a geometric path,
homotopy, continuous deformation of maps, product topology, equivalence of
paths under homotopy, fixed-end-point (FEP) homotopy, concatenation of paths,
constant path, binary operation, associative binary operation, identity element for
a binary operation, inverse of an element under a binary operation, first fundamental group, topological invariant

AlgTop24: The fundamental group

This lecture introduces the fundamental group of a surface. We begin by discussing when two paths on a surface are homotopic, then defining multiplication of paths, and then multiplication of equivalence classes or types of loops based at a fixed point of the surface.
The fundamental groups of the disk and circle are described.
This is part of a beginner's course on Algebraic Topology given by N J Wildberger at UNSW.
A screenshot PDF which includes AlgTop21 to 29 can be found at my WildEgg website here: http://www.wildegg.com/store/p119/product-Algebraic-Topology-C-screenshot-pdf

18:16

Who cares about topology? (Inscribed rectangle problem)

An unsolved conjecture, the inscribed square problem, and a clever topological solution to...

Who cares about topology? (Inscribed rectangle problem)

An unsolved conjecture, the inscribed square problem, and a clever topological solution to a weaker version of the question, the inscribed rectangle problem (Proof due to H. Vaughan, 1977), that shows how the torus and mobius strip naturally arise in mathematical ponderings.
Patreon: https://www.patreon.com/3blue1brown
Get 10% your domain name purchace from https://www.hover.com/, by using the promo code TOPOLOGY.
Special shout out to the following patrons: Dave Nicponski, Juan Batiz-Benet, Loo Yu Jun, Tom, Othman Alikhan, Markus Persson, Joseph John Cox, Achille Brighton, Kirk Werklund, Luc Ritchie, RiptaPasay, PatrickJMT , Felipe Diniz, Chris, Andrew Mcnab, Matt Parlmer, NaokiOrai, Dan Davison, JoseOscar Mur-Miranda, Aidan Boneham, BrentKennedy, Henry Reich, Sean Bibby, PaulConstantine, Justin Clark, Mohannad Elhamod, Denis, Ben Granger, Ali Yahya, Jeffrey Herman, and Jacob Young
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC-
Various social media stuffs:
Twitter: https://twitter.com/3Blue1Brown
Facebook: https://www.facebook.com/3blue1brown/
Reddit: https://www.reddit.com/r/3Blue1Brown

I will explain a viewpoint on resurgence via Floer-type topology of Lefschetz thimbles in the complexified path integral. In particular, so called 2d-4d wall-crossing formalism of Gaiotto, Moore and Neitzke arises naturally in this framework, without any reference to supersymmetric field theories.

7:54

How to use Path Deform and Topology in 3dsMax

A short video tutorial how to use Path Deform and Topology in 3dsMax.
You can download ex...

Jose Bravo demonstrates how to navigate through the QRM topology. Next he demonstrates the Configuration Monitor's ability to display the configurations of the network firewalls and routers in a vendor neutral way and to maintain a historical record of the configuration changes. He demonstrates how to highlight the differences between two sets of firewall configurations. He also demonstrate's QRM's ability to compare configurations between two different firewalls or any two devices.
Next, Jose demonstrates how QRM discovers new devices in the environment and he demonstrates how to search for devices in the topology so that missing machines can be located.
Jose also demonstrates QRM's capability to identify the path routes between two different machines and how the path between the two machines may be blocked.
IBM Owner: CalvinPowers

Recording of JonathanWilliamson's talk at the 2012BlenderConference on how to work with complex topology on organic and hard-surface models.
The video recording is pretty rough, the lighting in the room made it very difficult to get good footage. Thanks to the kind user that filmed the talk for us!

5:39

Network Theory Topology

Follow along with the course eBook: https://goo.gl/hhChUL
See the full course: http://comp...

Network Theory Topology

Follow along with the course eBook: https://goo.gl/hhChUL
See the full course: http://complexitylabs.io/courses
We call the overall structure to a network its topology where topology simply means the way in which constituent parts are interrelated or arranged. To illustrate how network topology affects the system we look at a number of simple networks each containing the same amount of nodes but each having a different overall topology owing to the way they were connected with these topologies affecting different features to the network, such as the shortest path length or how we might control the flow of information on the network.
Twitter: https://goo.gl/Nu6Qap
Facebook: https://goo.gl/ggxGMT
LinkedIn:https://goo.gl/3v1vwF
Transcription Except:
In the previous module, we were discussing local network metrics that referred to a single node in a graph, asking how connected it was or how influential and central it was while also building up our basic vocabulary from graph theory. In the coming set of lectures we are going to be looking at global metrics that refer to the whole of graph. A we have previously noted networks are a very informal type of structure they often simple develop without any overall top down design. Someone builds a protocol for two computers to exchange information over a network and shares it with a colleague, other people see the utility of it and connects to this little networks and then more people as the network grow, until 25 years later we have a massive network of networks that is the internet.
No one planned the internet just as one really planed the global financial networks that have emerged over the past few decades, traders, investor and institutions set up connections wherever they thought there was a viable return on investment, but now that these networks are here there overall markup feeds back to effect us the user, networks may start out quite random but they often develop into some stable overall structure and understanding the patterns to this overall structure is of central importance in network theory and the focus of this section to the course.
We can call this overall structure to a network its topology, where topology simply means the way in which constituent parts are interrelated or arranged:
Within the context of a network it defines the way different nodes are placed and interconnected with each other and the overall patterns that emerge out of this.
To illustrate this further here are a set of simple networks each containing the same amount of nodes but each having a different overall topology owing to the way it is connected. We can see how these different network topologies would in turn have very different features and properties to them. Imagine they where different transportation systems in which you are trying to get form A to B. In the star topology it would only ever take you two hops to reach your destination but in the ring it might take 3, the tree structure possibly 4, and this same influence from the topology would apply if we were trying to rout water through a hydraulic network or electricity on a power grid.
This network could also represent the flow of information within a national society we can see how the centralized star structure or the tree structure would be much easier for some political regime to control and influence as opposed to the more decentralized fully connected or ring network. The point I are trying to make clear is simply that networks have overall structure and this overall topology to the network matters as it feedback to affect the actions and capabilities of the nodes on the local level.

12:16

Graph Topological Sort Using Depth-First Search

In this video tutorial, you will learn how to do a topological sort on a directed acyclic ...

Graph Topological Sort Using Depth-First Search

In this video tutorial, you will learn how to do a topological sort on a directed acyclic graph (DAG), i.e. arrange vertices in a sequence according to dependency constraints shown by edges. The sort is done by using the depth-first search algorithm as the basis, and modifying it to label vertices with sequence numbers. The complete code is available at https://www.dropbox.com/s/71fe4ttpj27jxa8/Graph.java?dl=0
and the examples graphs illustrated in the video are at https://www.dropbox.com/s/6s8i9ji1wla28we/topsort_graph1.txt?dl=0
and
https://www.dropbox.com/s/tasix4uihrs1n5p/topsort_graph2.txt?dl=0
Error: At around 6:24, the graph example used has a cycle including X, C, D, P. The edge from X to C should go in the opposite direction, from C to X. Everything else works as it should.

Mod-02 Lec-08 Homotopy and the First Fundamental Group

An Introduction toRiemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T.E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
Goals of the Lecture:
- To understand the notion of homotopy of paths in a topological space
- To understand concatenation of paths in a topological space
- To sketch how the set of fixed-end-point (FEP) homotopy classes of loops at a point
becomes a group under concatenation, called the First FundamentalGroup
- To look at examples of fundamental groups of some common topological spaces
- To realise that the fundamental group is an algebraic invariant of topological spaces which
helps in distinguishing non-isomorphic topological spaces
- To realise that a first classification of Riemann surfaces can be done based on their fundamental
groups by appealing to the theory of covering spaces
Keywords:
Path or arc in a topological space, initial or starting point and terminal or ending point of a path,
path as a map, geometric path, parametrisation of a geometric path,
homotopy, continuous deformation of maps, product topology, equivalence of
paths under homotopy, fixed-end-point (FEP) homotopy, concatenation of paths,
constant path, binary operation, associative binary operation, identity element for
a binary operation, inverse of an element under a binary operation, first fundamental group, topological invariant

43:05

AlgTop24: The fundamental group

This lecture introduces the fundamental group of a surface. We begin by discussing when tw...

AlgTop24: The fundamental group

This lecture introduces the fundamental group of a surface. We begin by discussing when two paths on a surface are homotopic, then defining multiplication of paths, and then multiplication of equivalence classes or types of loops based at a fixed point of the surface.
The fundamental groups of the disk and circle are described.
This is part of a beginner's course on Algebraic Topology given by N J Wildberger at UNSW.
A screenshot PDF which includes AlgTop21 to 29 can be found at my WildEgg website here: http://www.wildegg.com/store/p119/product-Algebraic-Topology-C-screenshot-pdf

I will explain a viewpoint on resurgence via Floer-type topology of Lefschetz thimbles in the complexified path integral. In particular, so called 2d-4d wall-crossing formalism of Gaiotto, Moore and Neitzke arises naturally in this framework, without any reference to supersymmetric field theories.

The lecture was held within the framework of the Hausdorff Trimester Program : Applied and Computational Algebraic TopologyConcurrency theory in Computer Science studies effects that arise when several processors run simultaneously sharing common resources. It attempts to advise methods dealing with the "state space explosion problem" characterized by an exponentially growing number of execution paths; sometimes using models with a combinatorial/topological flavor. It is a common feature of these models that an execution corresponds to a directed path (d-path) in a (time-flow directed) state space, and that d-homotopies (preserving the directions) have equivalent computations as a result.
Getting to grips with the effects of the non-reversible time-flow is essential, and one needs to "twist" methods from ordinary algebraic topology in order to make them applicable. An essential task consists in inferring information about spaces of executions (directed paths) between two given states from information about the state space. The determination of path components is particularly important for applications.
I will discuss particular directed spaces arising from Higher Dimensional Automata (HDA). There are various methods identifying the homotopy type of the space of executions between two states in such an automaton with some finite complex: in simple cases as prodsimplicial complex – with products of simplices as building blocks – or as a configuration space living in a product of simplices. In several interesting cases, it is possible to give an explicit description of the homotopy type of the Alexander dual of such a configuration space and hence of the stable homotopy type of the corresponding trace space. This opens up for calculations of homology groups and of other topological invariants of some execution spaces.
We sketch a method recently devised by Ziemiański identifying – for a general HDA – a space of directed paths with a prodpermutahedral complex arising by glueing various permutahedra along their boundaries.
Joint work with L. Fajstrup (Aalborg), E. Goubault, E. Haucourt, S. Mimram (Éc. Polytechnique, Paris), R. Meshulam (Haifa) and K. Ziemiański (Warsaw).

AlgTop28: Covering spaces and fundamental groups

We illustrate the ideas from the last lectures by giving some more examples of covering spaces: of the torus, and the two-holed torus. Then we begin to explore the relationship between the fundamental groups of a covering space X and a base space B under a covering map p:X to B.
For this we need two important Lemmas: the Lifting Path lemma, and the Lifting Homotopy lemma. Then we obtain the basic result that the fundamental group of X can be viewed as a subgroup of the fundamental group of the base B, via the induced homomorphism of p. So the possibility emerges of studying covering spaces of B by studying subgroups of the fundamental group of B.
A screenshot PDF which includes AlgTop21 to 29 can be found at my WildEgg website here: http://www.wildegg.com/store/p119/product-Algebraic-Topology-C-screenshot-pdf

EE 101/10 - Topology, nodes, branches, series, parallel connections.

Basic as it sounds, the definitions in this short video are the linchpin of everything that follows. The bullet proof definitions will provide us with the necessary platform to analyze the most complicated electronic circuits that technology may throw at us.

39:00

What is a Manifold? Lesson 5: Compactness, Connectedness, and Topological Properties

The last lesson covering the topological prep-work required before we begin the discussion...

What is a Manifold? Lesson 5: Compactness, Connectedness, and Topological Properties

The last lesson covering the topological prep-work required before we begin the discussion of manifolds. Topics covered: compactness, connectedness, and the relationship between homeomorphisms and topological properties.

Using Blender's Light Path Node for Cycles

Learn how to use the LightPath node for Cycleshttp://cgcookie.com/blender/2013/02/26/blender-cycles-light-path-node/
Blender's light path node is incredibly useful for all sorts of tasks, but it's also a bit difficult to grasp at times and so many of us are left with no understanding of how to use it. This tutorial gives you an introduction to the light path node, showing you what can be done with it in Cycles. However, it's easier to understand how to use this tool when we know how render engines work, so we will give you a basic introduction to raytracing. You'll also learn why it works, and what you need to know in order to use it effectively.
We will be taking a look at how to control the influence of created materials on other objects in our scene. This includes changing the color of an object inside a reflection, in the shadow, through refraction, or any number of other ways. It also covers how to hide objects from specific light paths, such that it appears invisible in reflections, doesn't cast shadows, etc.

35:13

Introduction to the fundamental group

A leisurely introduction to the idea of homotopy, including a sketch proof that the fundam...

Introduction to the fundamental group

A leisurely introduction to the idea of homotopy, including a sketch proof that the fundamental group of the circle is the integers. This is aimed at second year undergrads embarking on projects in geometry/topology who need to quickly learn the basic idea of the fundamental group, so the focus is on developing intuition and explaining the definitions.

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Martin Raussen: Topological and combinatorial mode...

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Configuration Spaces and Topology of Robot Motion ...

AlgTop28: Covering spaces and fundamental groups...

PiTP 2015 - "Introduction to Topological and Confo...

EE 101/10 - Topology, nodes, branches, series, par...

What is a Manifold? Lesson 5: Compactness, Connect...

PiTP 2015 - "Introduction to Topological and Confo...

P. Friz - Rough paths, IHP 06/10/2014 - Part 1...

Using Blender's Light Path Node for Cycles...

Introduction to the fundamental group...

Point-Set Topology 7: Continuous Mappings & Interm...

Lecture - 11 The Graph Theory Approach for Electri...

14. Depth-First Search (DFS), Topological Sort...

Undergraduate Topology: Feb 7, connectedness (part...

Blender Conference 2012 - Jonathan Williamson: Wor...

Topology of Random Simplicial Complexes - Matthew ...

Gizmodo reported on Wednesday that a former Google engineer is suing the company for discrimination, harassment, retaliation, and wrongful termination ...Chevalier's posts had been quoting in Damore's lawsuit against Google, who is also suing the company for alleged discrimination against conservative white men ... “Firing the employee who pushed back against the bullies was exactly the wrong step to take.” ... But the effect is the same....

OSLO. Sea levels will rise between 0.7 and 1.2 metres in the next two centuries even if governments end the fossil fuel era as promised under the Paris climate agreement, scientists said on Tuesday ...Ocean levels will rise inexorably because heat-trapping industrial gases already em­­itted will linger in the atmosphere, melting more ice, it said. In addition, water naturally expands as it warms above four degrees Celsius (39.2F) ... ....

The woman tasked with caring for accused Florida shooter Nikolas Cruz and his brother have moved quickly to file court papers seeking control of their inheritance the day after the massacre at Majory Stoneman Douglas High School, Newsweek reported. When the mother of Nikolas and Zachary Cruz died from flu-related pneumonia last November, their lives were entrusted to Roxanne Deschamps, the report said....

Special CounselRobert Mueller's probe is prepared to accept a guilty plea from the London-based son-in-law of a Russian businessman after he made false statements during the investigation into alleged Russian interference in the 2016 U.S. presidential election, according to the Washington Post... Tymoshenko was later imprisoned by former president Viktor Yanukovych after signing a controversial deal with Russia for natural gas ... U.S ... U.S....

Article by WN.Com Correspondent Dallas DarlingTo this day it’s something my aunt hardly mentions, let alone discusses. And like a few other families living in the United States, it’s taboo and completely off limits ... Neither was it as widespread, since Japan had nearly conquered most of East Asia including parts of China. But still, U.S ... authorities continued the comfort station system absent formal slavery ... The U.S ... military authorities ... ....

Addressing the special plenary on pharmaceutical and biotechnology on the first day of UP Investors Summit, Patel said, "The summit will pave the path for development in the state ... and biotechnology on the first day of UP Investors Summit, Patel said, "The summit will pave the path for development in the state....

Michigan State PoliceTrooperPatrick Arena is being credited with saving English’s life after he used a tourniquet on the boy at the scene – but the path to recovery was just starting as he arrived at hospital and was put into a medically induced coma to aid in his healing. Today marks a milestone in the teen’s path to ......

MINNEAPOLIS (WCCO) — The Gopher men’s basketball team closes out its home portion of the schedule Wednesday night against Iowa. There’s no talk of NCAA Tournament positioning, just another Big Ten game, as both teams play out the season ... “I read a quote, like, two days ago that said, ‘If the path you take is easy, you’re on the wrong path,'” he said ... Mike MaxFacebookFollow on Twitter ... Comments ....

They custom-designed safety features – including a metal, flat-plate heatsink on the smartphone to dissipate excess heat from the laser, as well as a reflector-based mechanism to shut off the laser if a person tries to move in the charging beam’s path....

They custom-designed safety features – including a metal, flat-plate heatsink on the smartphone to dissipate excess heat from the laser, as well as a reflector-based mechanism to shut off the laser if a person tries to move in the charging beams path....

JakeEaster, who often rides past the UNC School of the Arts on his biking commute, said he often hits downhill speeds of around 35 miles an hour, but worries that someone getting out of a parked car could open a door into his path... “I know people that has happened to.” ... “I enjoy having an input into the process,” she said, adding that increasing the connectivity of biking paths is important as well ... ....

I think every rock and roll fan breathed a sigh of relief when Kid Rock decided not to run for political office. People should do what they are best at. Rock’s career path is hip-hop, metal, country and rock, in various combinations.So when the self-trained rapper/musician comes to the Wells Fargo Center, at 3601 S. Broad St., Philadelphia, Friday night, audience members can simply enjoy the music ... ....